Hopf surfaces in locally conformally Kähler manifolds with potential
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An LCK manifold with potential is a quotient M of a Kähler manifold X equipped with a positive plurisubharmonic function f, such that the monodromy group acts on X by holomorphic homotheties and maps f to a function proportional to f. It is known that a compact M admits an LCK potential if and only if it can be holomorphically embedded to a Hopf manifold. We prove that any non-Vaisman, compact LCK manifold with potential contains a complex surface (possibly singular) with normalization biholomorphic to a Hopf surface H. Moreover, H can be chosen non-diagonal, hence, also not admitting a Vaisman structure.
KeywordsLocally conformally Kähler Potential Hopf manifold Vaisman manifold
2000 Mathematics Subject Classification53C55
L.O. thanks the Laboratory for Algebraic Geometry at the Higher School of Economics in Moscow for hospitality and excellent research environment during February and April 2014, and April 2015. Both authors are indebted to Paul Gauduchon, Andrei Moroianu, and Victor Vuletescu for extremely useful disussions.
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